Significant inference and confidence sets for graphical models

Abstract: The problem of identifying statistically significant inferences about the structure of the graphical model is considered, along with the related task of constructing a confidence set for a graphical model. It has been proven that the procedure for constructing such set is equivalent to the procedure for simultaneous testing of hypotheses and alternatives regarding the composition of the graphical model. Some variants of the simultaneous testing of hypotheses and alternatives are discussed. It is shown that under the condition of free combination of hypotheses and alternatives, a simple generalization of the closure method leads to singlestep procedures for simultaneous testing of hypotheses and alternatives. The structure of the confidence set for the graphical model is analyzed, demonstrating how the confidence set leads to a separation of inferences about the graphical model into statistically significant and insignificant categories, or into an area of uncertainty. General results are detailed by analyzing confidence sets for undirected Gaussian graphical model selection. Examples are provided that illustrate the separation of inferences about the composition of undirected Gaussian graphical models into significant results and areas of uncertainty, and a comparison is made with known results obtained using the SINful approach to undirected Gaussian graphical model selection.